A polynomial $f\in \complex[z]$ defines a vector field $N_f(z) = -f(z)/f'(z)$ on $\complex$. Certain degenerate curves of flow in $N_f$ give the edges of the Newtonian graph, as defined by \cite{Sma85}. These give a relation between the roots of $f$ and $f'$, much similar to the linear order, when $f$ has real roots only. We give a purely algebraic algorithm to compute the Newtonian graph and the basins of attraction in the Newtonian field. The resulting structure can be used to query whether two points in $\complex$ are within the same basin of attraction. This gives us an algebraic approach to root-finding using Newton's method. This method extends to rational functions and more generally to any functions on $\complex$ whose flow is algebraic over $\complex(e)$.